Question

${\int }_1^3(x-[x\left]\right)dx$ is equal to (where $[.]$ denotes greatest integer function)

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is A $0$

$I={\int }_1^3(x-\left[x\right])dx={\int }_1^2(x-1)dx+{\int }_2^3(x-2)dx$

$={[\frac{x^2}{2}-x]}_1^2+{[\frac{x^2}{2}-2x]}_2^3$

$=[(2-2)-(\frac{1}{2}-1)]+[\frac{9}{2}-6-(2-4)]$

$=\frac{-1}{2}+\frac{9}{2}-6+2$

$\Rightarrow I=0$

Hence, the answer is $0.$